# Properties

 Degree 4 Conductor $2^{2} \cdot 5^{2} \cdot 41$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 4-s + 2·5-s + 2·9-s − 6·11-s + 16-s + 2·19-s − 2·20-s − 25-s + 2·29-s − 12·31-s − 2·36-s + 5·41-s + 6·44-s + 4·45-s − 6·49-s − 12·55-s + 4·59-s − 8·61-s − 64-s − 2·76-s + 12·79-s + 2·80-s − 5·81-s + 24·89-s + 4·95-s − 12·99-s + 100-s + ⋯
 L(s)  = 1 − 1/2·4-s + 0.894·5-s + 2/3·9-s − 1.80·11-s + 1/4·16-s + 0.458·19-s − 0.447·20-s − 1/5·25-s + 0.371·29-s − 2.15·31-s − 1/3·36-s + 0.780·41-s + 0.904·44-s + 0.596·45-s − 6/7·49-s − 1.61·55-s + 0.520·59-s − 1.02·61-s − 1/8·64-s − 0.229·76-s + 1.35·79-s + 0.223·80-s − 5/9·81-s + 2.54·89-s + 0.410·95-s − 1.20·99-s + 1/10·100-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$4100$$    =    $$2^{2} \cdot 5^{2} \cdot 41$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4100} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 4100,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.7902602434$ $L(\frac12)$ $\approx$ $0.7902602434$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;41\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;41\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 + T^{2}$$
5$C_2$ $$1 - 2 T + p T^{2}$$
41$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 6 T + p T^{2} )$$
good3$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
7$V_4$ $$1 + 6 T^{2} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$V_4$ $$1 - 22 T^{2} + p^{2} T^{4}$$
17$V_4$ $$1 + 22 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$V_4$ $$1 + 2 T^{2} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$V_4$ $$1 + 26 T^{2} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$V_4$ $$1 - 58 T^{2} + p^{2} T^{4}$$
53$V_4$ $$1 - 46 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 8 T + p T^{2} )$$
67$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$V_4$ $$1 + 74 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} )$$
97$V_4$ $$1 + 110 T^{2} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}